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vectormath_common.h
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vectormath_common.h
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/*************************** vectormath_common.h ****************************
* Author: Agner Fog
* Date created: 2014-04-18
* Last modified: 2019-08-01
* Version: 1.40.00
* Project: vector class library
* Description:
* Header file containing common code for inline version of mathematical functions.
*
* Theory, methods and inspiration based partially on these sources:
* > Moshier, Stephen Lloyd Baluk: Methods and programs for mathematical functions.
* Ellis Horwood, 1989.
* > VDT library developed on CERN by Danilo Piparo, Thomas Hauth and Vincenzo Innocente,
* 2012, https://root.cern.ch/doc/v606_/md_math_vdt_ReadMe.html
* > Cephes math library by Stephen L. Moshier 1992,
* http://www.netlib.org/cephes/
*
* Calculation methods:
* Some functions are using Pade approximations f(x) = P(x)/Q(x)
* Most single precision functions are using Taylor expansions
*
* For detailed instructions, see vcl_manual.pdf
*
* (c) Copyright 2014-2019 Agner Fog.
* Apache License version 2.0 or later.
******************************************************************************/
#ifndef VECTORMATH_COMMON_H
#define VECTORMATH_COMMON_H 1
#ifdef VECTORMATH_LIB_H
#error conflicting header files. More than one implementation of mathematical functions included
#endif
#include <math.h>
#include "vectorclass.h"
/******************************************************************************
define mathematical constants
******************************************************************************/
#define VM_PI 3.14159265358979323846 // pi
#define VM_PI_2 1.57079632679489661923 // pi / 2
#define VM_PI_4 0.785398163397448309616 // pi / 4
#define VM_SQRT2 1.41421356237309504880 // sqrt(2)
#define VM_LOG2E 1.44269504088896340736 // 1/log(2)
#define VM_LOG10E 0.434294481903251827651 // 1/log(10)
#define VM_LOG210 3.321928094887362347808 // log2(10)
#define VM_LN2 0.693147180559945309417 // log(2)
#define VM_LN10 2.30258509299404568402 // log(10)
#define VM_SMALLEST_NORMAL 2.2250738585072014E-308 // smallest normal number, double
#define VM_SMALLEST_NORMALF 1.17549435E-38f // smallest normal number, float
#ifdef VCL_NAMESPACE
namespace VCL_NAMESPACE {
#endif
/******************************************************************************
templates for producing infinite and nan in desired vector type
******************************************************************************/
template <class VTYPE>
static inline VTYPE infinite_vec();
template <>
inline Vec2d infinite_vec<Vec2d>() {
return infinite2d();
}
template <>
inline Vec4f infinite_vec<Vec4f>() {
return infinite4f();
}
#if MAX_VECTOR_SIZE >= 256
template <>
inline Vec4d infinite_vec<Vec4d>() {
return infinite4d();
}
template <>
inline Vec8f infinite_vec<Vec8f>() {
return infinite8f();
}
#endif // MAX_VECTOR_SIZE >= 256
#if MAX_VECTOR_SIZE >= 512
template <>
inline Vec8d infinite_vec<Vec8d>() {
return infinite8d();
}
template <>
inline Vec16f infinite_vec<Vec16f>() {
return infinite16f();
}
#endif // MAX_VECTOR_SIZE >= 512
// Define NAN paload values
#define NAN_LOG 0x101 // logarithm for x<0
#define NAN_POW 0x102 // negative number raised to non-integer power
#define NAN_HYP 0x104 // acosh for x<1 and atanh for abs(x)>1
/******************************************************************************
templates for polynomials
Using Estrin's scheme to make shorter dependency chains and use FMA, starting
longest dependency chains first.
******************************************************************************/
// template <typedef VECTYPE, typedef CTYPE>
template <class VTYPE, class CTYPE>
static inline VTYPE polynomial_2(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2) {
// calculates polynomial c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
//return = x2 * c2 + (x * c1 + c0);
return mul_add(x2, c2, mul_add(x, c1, c0));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_3(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3) {
// calculates polynomial c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
//return (c2 + c3*x)*x2 + (c1*x + c0);
return mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_4(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4) {
// calculates polynomial c4*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
//return (c2+c3*x)*x2 + ((c0+c1*x) + c4*x4);
return mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0) + c4*x4);
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_4n(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3) {
// calculates polynomial 1*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
//return (c2+c3*x)*x2 + ((c0+c1*x) + x4);
return mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0) + x4);
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_5(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4, CTYPE c5) {
// calculates polynomial c5*x^5 + c4*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
//return (c2+c3*x)*x2 + ((c4+c5*x)*x4 + (c0+c1*x));
return mul_add(mul_add(c3, x, c2), x2, mul_add(mul_add(c5, x, c4), x4, mul_add(c1, x, c0)));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_5n(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4) {
// calculates polynomial 1*x^5 + c4*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
//return (c2+c3*x)*x2 + ((c4+x)*x4 + (c0+c1*x));
return mul_add(mul_add(c3, x, c2), x2, mul_add(c4 + x, x4, mul_add(c1, x, c0)));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_6(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4, CTYPE c5, CTYPE c6) {
// calculates polynomial c6*x^6 + c5*x^5 + c4*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
//return (c4+c5*x+c6*x2)*x4 + ((c2+c3*x)*x2 + (c0+c1*x));
return mul_add(mul_add(c6, x2, mul_add(c5, x, c4)), x4, mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0)));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_6n(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4, CTYPE c5) {
// calculates polynomial 1*x^6 + c5*x^5 + c4*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
//return (c4+c5*x+x2)*x4 + ((c2+c3*x)*x2 + (c0+c1*x));
return mul_add(mul_add(c5, x, c4 + x2), x4, mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0)));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_7(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4, CTYPE c5, CTYPE c6, CTYPE c7) {
// calculates polynomial c7*x^7 + c6*x^6 + c5*x^5 + c4*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
//return ((c6+c7*x)*x2 + (c4+c5*x))*x4 + ((c2+c3*x)*x2 + (c0+c1*x));
return mul_add(mul_add(mul_add(c7, x, c6), x2, mul_add(c5, x, c4)), x4, mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0)));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_8(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4, CTYPE c5, CTYPE c6, CTYPE c7, CTYPE c8) {
// calculates polynomial c8*x^8 + c7*x^7 + c6*x^6 + c5*x^5 + c4*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
VTYPE x8 = x4 * x4;
//return ((c6+c7*x)*x2 + (c4+c5*x))*x4 + (c8*x8 + (c2+c3*x)*x2 + (c0+c1*x));
return mul_add(mul_add(mul_add(c7, x, c6), x2, mul_add(c5, x, c4)), x4,
mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0) + c8*x8));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_9(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4, CTYPE c5, CTYPE c6, CTYPE c7, CTYPE c8, CTYPE c9) {
// calculates polynomial c9*x^9 + c8*x^8 + c7*x^7 + c6*x^6 + c5*x^5 + c4*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
VTYPE x8 = x4 * x4;
//return (((c6+c7*x)*x2 + (c4+c5*x))*x4 + (c8+c9*x)*x8) + ((c2+c3*x)*x2 + (c0+c1*x));
return mul_add(mul_add(c9, x, c8), x8, mul_add(
mul_add(mul_add(c7, x, c6), x2, mul_add(c5, x, c4)), x4,
mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0))));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_10(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4, CTYPE c5, CTYPE c6, CTYPE c7, CTYPE c8, CTYPE c9, CTYPE c10) {
// calculates polynomial c10*x^10 + c9*x^9 + c8*x^8 + c7*x^7 + c6*x^6 + c5*x^5 + c4*x^4 + c3*x^3 + c2*x^2 + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
VTYPE x8 = x4 * x4;
//return (((c6+c7*x)*x2 + (c4+c5*x))*x4 + (c8+c9*x+c10*x2)*x8) + ((c2+c3*x)*x2 + (c0+c1*x));
return mul_add(mul_add(x2, c10, mul_add(c9, x, c8)), x8,
mul_add(mul_add(mul_add(c7, x, c6), x2, mul_add(c5, x, c4)), x4,
mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0))));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_13(VTYPE const & x, CTYPE c0, CTYPE c1, CTYPE c2, CTYPE c3, CTYPE c4, CTYPE c5, CTYPE c6, CTYPE c7, CTYPE c8, CTYPE c9, CTYPE c10, CTYPE c11, CTYPE c12, CTYPE c13) {
// calculates polynomial c13*x^13 + c12*x^12 + ... + c1*x + c0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
VTYPE x8 = x4 * x4;
return mul_add(
mul_add(
mul_add(c13, x, c12), x4,
mul_add(mul_add(c11, x, c10), x2, mul_add(c9, x, c8))), x8,
mul_add(
mul_add(mul_add(c7, x, c6), x2, mul_add(c5, x, c4)), x4,
mul_add(mul_add(c3, x, c2), x2, mul_add(c1, x, c0))));
}
template<class VTYPE, class CTYPE>
static inline VTYPE polynomial_13m(VTYPE const & x, CTYPE c2, CTYPE c3, CTYPE c4, CTYPE c5, CTYPE c6, CTYPE c7, CTYPE c8, CTYPE c9, CTYPE c10, CTYPE c11, CTYPE c12, CTYPE c13) {
// calculates polynomial c13*x^13 + c12*x^12 + ... + x + 0
// VTYPE may be a vector type, CTYPE is a scalar type
VTYPE x2 = x * x;
VTYPE x4 = x2 * x2;
VTYPE x8 = x4 * x4;
// return ((c8+c9*x) + (c10+c11*x)*x2 + (c12+c13*x)*x4)*x8 + (((c6+c7*x)*x2 + (c4+c5*x))*x4 + ((c2+c3*x)*x2 + x));
return mul_add(
mul_add(mul_add(c13, x, c12), x4, mul_add(mul_add(c11, x, c10), x2, mul_add(c9, x, c8))), x8,
mul_add(mul_add(mul_add(c7, x, c6), x2, mul_add(c5, x, c4)), x4, mul_add(mul_add(c3, x, c2), x2, x)));
}
#ifdef VCL_NAMESPACE
}
#endif
#endif